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Error propagation: Difference between revisions
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This technique also provides a quick method to check for possible problems such as ill-conditioning, cited above. | This technique also provides a quick method to check for possible problems such as ill-conditioning, cited above. | ||
When the model relating ''s'' and ''p'' is known, as well as the error distributions (and the latter may either be Gaussian or not), a more systematic approach to error propagation is provided by a technique known as the [[:Wikipedia:Maximum likelihood|maximum likelihood method]]. | When the model relating ''s'' and ''p'' is known, as well as the error distributions (and the latter may either be Gaussian or not), a more systematic approach to error propagation is provided by a technique known as the [[:Wikipedia:Maximum likelihood|maximum likelihood method]]. | ||
<ref>Particle Data Group, Eur. Phys. J. C 3, 1 (1998)</ref> | <ref>[http://dx.doi.org/10.1007/s10052-998-0104-x Particle Data Group, Eur. Phys. J. C 3, 1 (1998)]</ref> | ||
This technique is simply the generalisation of standard error propagation to general error distributions (i.e. not limited to Gaussians). | This technique is simply the generalisation of standard error propagation to general error distributions (i.e. not limited to Gaussians). | ||
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<ref>J. van den Berg, ed., Wavelets in Physics (Cambridge University Press, 1999) ISBN 978-0521593113</ref> | <ref>J. van den Berg, ed., Wavelets in Physics (Cambridge University Press, 1999) ISBN 978-0521593113</ref> | ||
determination of fractal dimension, mutual information, reconstruction of chaotic attractor, | determination of fractal dimension, mutual information, reconstruction of chaotic attractor, | ||
<ref>H. Abarbanel, R. Brown, J. Sidorowich, and L. S. Tsimring, Rev. Mod. Phys. 65, 1331 (1993)</ref> ...). | <ref>[http://link.aps.org/doi/10.1103/RevModPhys.65.1331 H. Abarbanel, R. Brown, J. Sidorowich, and L. S. Tsimring, Rev. Mod. Phys. 65, 1331 (1993)]</ref> ...). | ||
== Non-Gaussian statistics == | == Non-Gaussian statistics == | ||
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The distribution of random variations of a signal ''s'' around its mean value need not be Gaussian. | The distribution of random variations of a signal ''s'' around its mean value need not be Gaussian. | ||
E.g., photon statistics are typically of the Poisson type, which is especially important for low signal levels. | E.g., photon statistics are typically of the Poisson type, which is especially important for low signal levels. | ||
<ref>B. van Milligen, I. Classen, and C. Barth, Rev. Sci. Instrum. 74, 3998 (2003)</ref> | <ref>[http://link.aip.org/link/?RSINAK/74/3998/1 B. van Milligen, I. Classen, and C. Barth, Rev. Sci. Instrum. 74, 3998 (2003)]</ref> | ||
In other cases, the random component of the signal ''s'' is simply a non-linear function of a (Gaussian) noise source, causing the distribution to be skewed or distorted. | In other cases, the random component of the signal ''s'' is simply a non-linear function of a (Gaussian) noise source, causing the distribution to be skewed or distorted. | ||
Or the random component of the measured signal could correspond to the maximum or minimum value of a (Gaussian) random number, leading to extremal (Gumbel) distributions. | Or the random component of the measured signal could correspond to the maximum or minimum value of a (Gaussian) random number, leading to extremal (Gumbel) distributions. | ||
<ref>B. van Milligen, R. Sánchez, B. Carreras, V. Lynch, B. LaBombard, M. Pedrosa, C. Hidalgo, B. Gonçalves, and R. Balbín, Phys. Plasmas 12, 052507 (2005)</ref> | <ref>[http://link.aip.org/link/?PHPAEN/12/052507/1 B. van Milligen, R. Sánchez, B. Carreras, V. Lynch, B. LaBombard, M. Pedrosa, C. Hidalgo, B. Gonçalves, and R. Balbín, Phys. Plasmas 12, 052507 (2005)]</ref> | ||
The log-normal distribution is also quite common (e.g. in potential fluctuations). | The log-normal distribution is also quite common (e.g. in potential fluctuations). | ||
<ref>F. Sattin, N. Vianello, and M. Valisa, Phys. Plasmas 11, 5032 (2004)</ref> | <ref>[http://link.aip.org/link/?PHPAEN/11/5032/1 F. Sattin, N. Vianello, and M. Valisa, Phys. Plasmas 11, 5032 (2004)]</ref> | ||
However, all the previous distributions can be obtained by suitable manipulations of Gaussian random variables. | However, all the previous distributions can be obtained by suitable manipulations of Gaussian random variables. | ||
A totally different class of statistics is known as Lévy distributions (of which the Gaussian distribution is only a special case), which is the class of distributions satisfying the requirement that the sum of independent random variables with a distribution ''P'' again has a distribution ''P'' (generalisation of the Central Limit Theorem). | A totally different class of statistics is known as Lévy distributions (of which the Gaussian distribution is only a special case), which is the class of distributions satisfying the requirement that the sum of independent random variables with a distribution ''P'' again has a distribution ''P'' (generalisation of the Central Limit Theorem). | ||
Such distributions are expected to appear in self-organised systems (such as plasmas). | Such distributions are expected to appear in self-organised systems (such as plasmas). | ||
In general, the detection of this type of non-Gaussian statistics is difficult. Some techniques are however available, such as renormalisation, rescaled-range analysis, | In general, the detection of this type of non-Gaussian statistics is difficult. Some techniques are however available, such as renormalisation, rescaled-range analysis, | ||
<ref>B. Carreras, B. van Milligen, M. Pedrosa, R. Balbín, C. Hidalgo, D. Newman, E. Sánchez, R. Bravenec, G. McKee, I. García-Cortés, et al., Phys. Plasmas 6, 1885 (1999)</ref> | <ref>[http://link.aip.org/link/?PHPAEN/6/1885/1 B. Carreras, B. van Milligen, M. Pedrosa, R. Balbín, C. Hidalgo, D. Newman, E. Sánchez, R. Bravenec, G. McKee, I. García-Cortés, et al., Phys. Plasmas 6, 1885 (1999)]</ref> | ||
the detection of long-range time dependence, | the detection of long-range time dependence, | ||
<ref>B. Carreras, D. Newman, B. van Milligen, and C. Hidalgo, Phys. Plasmas 6, 485 (1999)</ref> | <ref>[http://link.aip.org/link/?PHPAEN/6/485/1 B. Carreras, D. Newman, B. van Milligen, and C. Hidalgo, Phys. Plasmas 6, 485 (1999)]</ref> | ||
finite-size Lyapunov exponents, | finite-size Lyapunov exponents, | ||
<ref>B. Carreras, V. Lynch, and G. Zaslavski, Phys. Plasmas 8, 5096 (2001) </ref> etc. | <ref>[http://link.aip.org/link/?PHPAEN/8/5096/1 B. Carreras, V. Lynch, and G. Zaslavski, Phys. Plasmas 8, 5096 (2001)]</ref> etc. | ||
Sometimes it is possible to obtain information on the nature of the errors by averaging experimental data (in space or time) - this is the renormalisation technique referred to above. | Sometimes it is possible to obtain information on the nature of the errors by averaging experimental data (in space or time) - this is the renormalisation technique referred to above. | ||
When averaging over ''N'' samples, the variation of the ''N''-averaged (or smoothed) data is less than that of the original data. | When averaging over ''N'' samples, the variation of the ''N''-averaged (or smoothed) data is less than that of the original data. | ||